= Solution
Let $H:X\to Y$ be a <homeomorphism> with $H\circ f=g\circ H$. This is <topological conjugacy>. Choose a <compatible metric> $e$ on $Y$ and pull it back to $X$:
$$
d_H(x,x')=e(Hx,Hx').
$$
Because $H$ is a <homeomorphism>, $d_H$ is a <compatible metric>. The conjugacy identity implies $Hf^k=g^kH$ for every $k\geq0$, so the corresponding <Bowen metrics> satisfy
$$
(d_H)_n(x,x')=\max_{0\leq k<n}e(g^kHx,g^kHx')=e_n(Hx,Hx').
$$
The <bijection> $H$ therefore transports <separated sets> in either direction without changing their separation, and $s_n(d_H,\epsilon)=s_n(e,\epsilon)$. Their exponential growth rates, and then their limits as $\epsilon\downarrow0$, agree. Part (a) lets us replace $d_H$ by any <compatible metric> on $X$. Hence
$$
\boxed{h_{\mathrm{top}}(f)=h_{\mathrm{top}}(g).}
$$
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